Image depicting a Calabi-Yau manifold, origin unknown (please tell me if you know for credits).
Super-rigidity and transversality for multiple covers in SFT.
Nearby Lagrangian conjecture for Pinwheels, joing with N. Adaloglou and J. Hauber.
I have spent some time thinking about foundational aspects of (pseudo)holomorphic curves. These are holomorphic maps from Riemann surfaces to (almost)complex manifolds, their theory is rich: the local nature of their moduli is somehow controlled by their holomorphic nature (meaning rigidity and structure from algebraic geometry presented by an elliptic PDE) and their compactness properties are governed by their Riemannian nature (they are even better than stable minimal surfaces).
Why should you care about these objects? Here you have a short "survey" that displays several results I find cute and pretty. It is by no means comprehensive but I have tried to emphasize their versatility.
Amongst this foundational aspects, I have studied the new techniques in equivariant transversality introduced by Wendl in his super-rigidity paper. The idea is to quantify the way transversality (and rigidity) fail. Here is a soft "layperson" explanation of the contents and directions of this paper and super-rigidity. By layperson I mean someone acquainted with either differential or algebraic geometry but not much else, so an advanced undergrad or motivated undergrad should be able to follow the text just fine.